Figure REF  shows the curves of the derivative of
the phonon Hall conductivity and the Chern numbers at the
critical magnetic fields.
The first derivative of phonon Hall conductivity has a
minimum or maximum at the magnetic fields \(h_{c1}=5.07, h_{c2}=6.75,{\rm and} h_{c3}=20.39\)  rad/ps for the finite-size sample (the sample has
\(N=N_L^2\)  unit cells).
The first derivative \(d\kappa _{xy}/dh\)  at the points \(h_{c1},h_{c2}, h_{c3}\)  diverges when the system size increases to infinity
[1]}. At the three critical points the second derivative
\(d^2\kappa _{xy}/dh^2\)  is discontinuous, which is shown in the inset of
Fig. REF , across which phase transitions occur.
For different temperatures, the phase transitions occur at exactly the
same critical values. Thus the temperature-independent phase transition
does not come from the thermodynamic effect, but is induced by the topology
of the phonon band structure, which corresponds to the sudden change
of the Chern numbers. While there
is one discontinuity of the Chern numbers for the honeycomb lattice
system, for the kagome lattice system, there are three ones corresponding
to the divergency of the derivative of the phonon conductivity, which can
be seen in Fig. REF .
